 Okay, so I start from a real quadratic field. So I have our rational numbers and I have f that is q square root of b or something, b is a positive and then I have a parallel representation so I take no bar that is an induced representation from the Galangal 4 bar s to Galangal 4 bar q of a character 5 bar having values in our b field to define an extension of f close, but actually for simplicity I assume that f is just equal to fp so this is fp cross and then I have a parallel lift zp cross into it and that I write just 5 and so I have a splitting field of this f root bar so often I just write this induced representation, the ultimate representation corresponding to this 5 as rho so that the same splitting field it has and then I take fp of rho bar that is max p-profinite ramified only at p extension and I write the Galangal root from the top as g so the index 2 sub root here is h so I will do the deformation over this group g and h and I suppose actually I fix so the p is supposed to I'm a very ordinary mathematician so I love to be ordinary and therefore p is splitting into two pieces so the sigma is a Galangal element here but sigma acts non-trivial in this quadratic field and so there's no so I fix this guy and also I fix 1,000,000 of f into real number this is also fixed I just write conductor of this character 5 to be the gothic c and infinity and I suppose it is made of split plans so o is an integer ring and also I want to suppose that it is prime to p gothic p conjugate under this kind of circumstances I also it is about almost all plans splitting in this field f so I just assume that p doesn't divide the class number of f cross number of splitting field of phi minus phi minus is the antipsychotic projection so phi minus is phi, phi, sigma inverse phi, sigma tau phi, sigma tau, sigma inverse something like that so this is the kind of assumptions I make then I have a deformation ring so t and rho t is the universal guy so universal, ordinary garrot deformation ring with universal representation of rho t of this robot and the deformation I make over this loop g so it's fairly small okay so this is actually a HEC algebra and it's a local ring of the HEC algebra so it is algebra free of finite rank over the sub-argebra, weight in the sub-argebra so it's a variable t and I suppose that this is rank p over lambda okay the as you know, robot is induced representation so you pick a quadratic residue symbol then you tensor chi with rho, robot then it is isomorphic to robot as you know that characterizes induced representation and therefore if you make a t tensor product with universal guy it still remains to be a deformation of the robot so you have a bioniversality you have an automorphism iota of ring, universal ring t so the iota is active on it it's an involution, obviously, chi square is 1 and I usually write t plus for the sub-ring fixed by this iota okay so this is still for your finite rank half of e so e is an even number okay so what my goal of this talk my goal of this talk is just to prove prime fact the t is aomorphic to lambda square root of bracket epsilon minus 1 the epsilon is a fundamental unit so the plus minus e to the v is just all closed and I just write rho a case t 1 plus t the unit in lambda and the bracket epsilon is p to the rho of epsilon over rho p 1 plus p oh I forgot to say that p is bigger than or equals to 5 there's one more condition to have to have this kind of deformation ring I suppose that phi minus restricted to the decomposition group has all that bigger than 2 and globally phi bar has all that bigger than 3 this condition is technical in the sense that if it is 2 and if it is 1 you don't have a it's a reducible case and if it is 2 you could have write down this rho bar the reducible representation from other quadratic field and it makes things quite bad so I just assume this under this kind of assumption I try to show this and one application is that for example you take a serban group of a q of a joint representation of rho t this is for example green bar the serban group and you take this font trail in dual then this is isomorphic to lambda I call epsilon that is lambda modulo epsilon minus 1 something like that this is out of by a theorem of omega this is just a omega t over lambda and you know what it is you can compute this this is sort of major theorem so you get this so one remark is that so if you write gosic p order of epsilon to the p minus 1 minus 1 to be summated by m that is bigger than 1 then this idea of epsilon minus 1 is p to the p to the m minus 1 minus 1 this is divisible by capital p so it's all the time on trivial and another point is that this is psychotomy polynomial root is people so it's square free I think this is a unique case we know that the serban group characterization polynomial not just cyclic but it's semisimple so to speak no multiple root and this is another point is that this highly depends on the real quadratic field but it's not independent of this fiber character what weight is that corresponding weight what that that this inducer law you get okay so then I recall something I did quite ago that sort of the paper cyclicity of a joint serban group and fundamental rate I wrote a paper a couple years ago this is posted on the web on my home page so you can take a look and what I did in this paper is that the following things I have shown okay so it's I call presentation serban so DDT has a presentation of lambda one variable power 3 one more so it's a entirely two variable modular power series S and you can lift can lift the involution iota of T to an involution iota double block at X and the iota infinity at some X by minus iota infinity at some S by plus okay so actually S is a power series of T square X square alright so this is what I proved in that paper and this result would actually tell you that this is also the Selma group of Q of induced representation F so a joint of induced representation F over Q of universal character 5 5 is a chapter 5 universal character deforming the phi bar into Zp so the gamma group of this kind of thing is a low p cross you need to take a p part or p cross you model epsilon to the Z cross minus but I will anyway take Zp portion and that that is obviously lambda epsilon cross this is the universal character only lamified at half this gothic phi and half gothic phi also possibly and that's something like that and this theorem tells you that this is isomorphic to omega T over T plus tensor T over I what I is T over I is the maximal portion on phi I ought to act trivially so this is T modulo T I ought to minus 1P so that's the definition of I and on phi actually this is the universal induced representation so it has to be come from the universal character so this portion is lambda epsilon and the low T modulo this I is induced representation of capital phi okay so low T mod I is induced FQ capital phi okay so this is the setting I found normal not normal but normal okay so now I'm going back to good old days of German mathematics I just applied biostress preparation theorem because this is free of finite rank E over lambda so you can write down SX as a product of distinguished polynomial and a unit power series so this is biostress okay but on the other hand on the other hand by this presentation by this presentation X goes down to I think theta I call it and X square goes down down to theta square so I call it low ACA theta theta square so this comes from X square and obviously T is lambda generated by capital theta because of this presentation and this theta is the root of this distinguished polynomial okay so D has a degree E and T plus is there for lambda or low ACA theta something like that so I just like to say like this collapse into lambda alright then so the I is there for generated by capital theta right because theta is the element on which I have the arcs of minus one and therefore lambda double bracket X modulo X and S is low of the shadow oh there's a shadow oh okay fine okay so I'm quite in the shadow okay so this is lambda modulo S0 and this is on the other hand this is P over theta so this is lambda epsilon lambda epsilon minus one like that so S0 as you can assume just by adjusting by unit is just epsilon minus one as I remarked this is square free I plug in zero here P0 U0 is S0 so that is epsilon minus one therefore again adjusting S by unit this is the limit I can assume that V0 is epsilon minus one okay I can do this by preparation business over lambda P I think a prime capital P dividing this V0 so epsilon minus one height one prime so prime divisor I can make a localization and a completion and do the wire stress preparation theorem again under these circumstances I get at the composition of Sx into Dpx and UPx now this guy is a unit in lambda p hat double black attacks but as you know this is the morning distinguished it's unique for any way you make a wire stress preparation theorem so the D is equal to Dp for any capital P dividing epsilon minus one so it's a nice Eisenstein polynomial because this bracket epsilon minus one is square free okay so I get a following theorem so I get a following in the USVT theorem P is integral domain normal integral domain ramified every capital P dividing epsilon minus one and if P is two you get T is lambda essentially square root of epsilon minus one so I need to show this okay so that's something I try okay by the way as I said when this order I haven't erased anything I just wrote the order of this the epsilon to the P minus one order of basic P to the M and this kind of prime so M if M is kind of bigger than two this kind of prime is called all sansan polynomial it is something analytic number of theories sansan is a twin Chinese plus and they wanted to show counter example of Fermat's raciolum and if Fermat's raciolum fails they do it over a cure to square root of five Fermat's raciolum fails then it has to be also and so for analytic number of theories this is something slightly interesting and as you know as I said this epsilon bracket minus one is just capital P when M is one therefore under this setting P is regular factorial if one leave M is one otherwise we have ambiguous process and that is not more sansan prime ok and they they computed people computed this more sansan primes and yeah no more sansan primes but let me know the Fermat's raciolum is two so no more sansan left down because it's not a final leaf thing so even though Fermat's raciolum fails it could you could have a more sansan plan for example if you do this for cure square root of ten then that example P is 191 693 and then 60 something like that these analytic number is conjecture that there's perhaps density zero but infinitely many such points so this factorial property regularity comes from vanishing of certain H2 right so H2 vanishes most of the times that's kind of natural okay so that's kind of a lot and so I just really try to prove this equal to two I don't try to prove the length of T plus is one but I rather like to try to prove the ramification index is two at that induced point okay so I use see all you belay shenivier everybody knows I put them into the shadow as I told you that belay shenivier everybody know so I put it in the shadow alright and so you now out of the shadow you have you have a two character of dimension two deforming so I have I have five plus five inner conjugate sigma is absolute character residual alright so you have a deformation of this and I assume that the dimension two and two you have of course the trace and you have T three I don't like to write it down but it is a six term identity you have all possibility of this kind of product processing and I add one more thing is that that T I consider this kind of function T from your group of what so it from A is the ring and H into A is a commutative ring whose residue local whose residue field is Fp and that T is half of T of outside square inside square and this I suppose this is multiplicative and they sort of studied the strategy in general so this kind of should character may not lift to the real two dimension representation into gf2 of A but it would lift to sort of gma generalized matrix and geoblasso and generalized matrix geoblasso is something like A here B and C B C A modules and you you have a product B tensor AC into A and C tensor AB into A so that use your matrix products made to this A so that is called a generalized matrix algebra of course this product has to satisfy certain axiom to guarantee that is associativity and so on but that then recently so again the press on wake and our one Edison considered nearly sort of this kind of KD Hamilton representation KD Hamilton representation means that it's just a representation of say A h into this gma I call it E E cross such that it satisfies characteristic polynomial so R square minus T R R plus that T of R is zero something then we call it KD Hamilton okay and they studied so I put their names in the shadow the wake and then the universal pair are ordinary and E ordinary this is R ordinary B ordinary C ordinary R ordinary something like that and then you have a universal representation R ordinary h into this E ordinary cross which satisfies obvious universal property here ordinary is that only the composition rule you have two things I mean these guys consider the one case and over Q but you can do it of course over any field and it has to be upper triangular for D Gossik P and lower triangular for D Gossik P Sigma and upper mirabolic over in Asia lower mirabolic this is my ordinary condition okay if you make it all upper triangular it's a different problem because it's a reducible case and I need to choose this way okay then so you have a universal guide and now by the right you have an idea of reducibility means that it is a minimal sort of idea and you take modular this J this becomes reducible in the sense that these two guys or diagonal guys becomes character okay and then obviously the product of this has to be zero right so therefore J is the actually the image of B or tensor C or over R in R then you get a universal reducible people R X that is R over J E old E lead that is E old over J and low red that is low old modular J and this is the universal among reducible guys okay then you know the character so this E old I just write down as a diagonal R red R red B red tensor product B red C red with the product map is a zero map but these guys could survive okay and the character I mean this character having this is the universal among among the character performing fine so therefore this character low one one red has to be fine low two two red has to be fine conjugate inside okay so and pass maybe up then if you have therefore this reducible universal guy and five five sigma and something here something there but this guy will govern the extension of five five sigma right therefore it is governed by from a girl group into this guy and that girl group here is something like you have F you have F five minus because this guy divided by that is five minus and then then you have a Gothic P only Gothic P only five and P sigma totally split okay that kind of girl group and this girl group I just want Y and then I just like Y five minus the five minus branch of this Y then it's kind of obvious you get a Y five minus here it's conjugate so sigma Y five minus sigma inverse is there so you get an explicit form of this reducible guy and one more thing is that as I said sigma group of a joint induce a joint induce of capital five is again lambda epsilon so this is this is like morphic to lambda epsilon so this is a result of my own paper I quoted okay so this is cyclic over lambda and cyclic therefore over alright okay so this is the same so it looks like two by two matrix of lambda epsilon but product is zero back so be careful about that then so I determine now the reducible quotient so I try now understand the original ordinary guy universal ordinary K Hamilton and so it goes something like that it's very easy to prove so you know you have this odd and you have a T like you have a deformation here you restrict it to H and take say trace right this you have therefore your mark because this is a universal object okay actually you can easily show that this factor through T plus and also easy to show that this is onto this I use this presentation that's important for me to show on to this but once you get a presentation it's trivial okay then as you know that's my claim is that this this map is a nice move okay so I just drew the picture I drew the dialogue but the point here is that our old tensor no be old be old you model out be old you get that be red and that is a cyclic so be old is cyclic over R old same C old is cyclic over R old therefore be old tensor C old over R old it's a depth down J but these two are the cyclic modules so this is cyclic okay so this is a principal idea okay so I draw the picture a diagram R old module this principal idea so this is a red as I told you this is lambda epsilon I have this map t plus you remember lambda t plus t t so I have ideal i generated by capital theta so it's intersection with t plus i plus it is generated by theta and this is fully labified and therefore t plus module i plus generated by theta is also lambda epsilon and it's all two so it is a nice morphism okay then because theta this lower theta by I know that is non-zero divisor therefore you multiply by theta to the n I get i plus n over i plus n plus lambda here you can do the same thing j or n j n plus 1 yeta to the n and by such activity this is subjective this is subjective this side because it's a non-zero divisor all isomorphism this upper corner so this bottom corner is also isomorphism I get jn over jn plus 1 it's isomorphic to i plus n i plus n plus 1 and therefore I get t plus it's isomorphic to r or this actually I conjectured in a very different form as in terms of L-parries during the L-parries long ago in a paper of Doe and Rishi in Benzionis so I sort of proved as rightfully this kind of time alright so the ordinary guy as I told you that it is r t r b r the modulo j is cyclic then by Nakayama this is cyclic so r r modulo some idea so this is same way here r r on the other hand I have this r t originally having values in g r to t but you restrict it in h then by conversion so the cositre calculation you do you can make it having values in t plus t minus that is theta times t plus t minus t plus it has values in so h goes into goes into this one this is of course obviously KB Hamilton representation alright and so this guy covers this by universality subjective and theta is non-zero device of real and quantity plus but I have proved that this is old so this map is isomorphic h0 so this is the universal graph so this is that's what I get to the end one more thing I need to show is that low t you restrict it in inertia group at p inside the galore group of f low t splitting field of this f low bar so this is a p-propinant graph and you can easily show actually my paper in Compositional 2015 I have basically shown that this has a form t to the zp 01 and some idea here non-zero non-zero come from the result of Joven so it's intercomposability local and the point here is that if this guy modulo maximum idea of t vanishes then this guy of course factors through y of 5 minus so it is un-lambified outside p right and it vanishes modulo maximum idea it's un-lambified everywhere but it's not trivial because factors look at one group so p divides hf 5 minus obviously but this is I suppose that p doesn't divide by 5 minus so what I'm saying is that this square is actually ck times y of 3 this is rank 1 y of 3 is a local y of 3 y of 3 is a prove in his famous paper of zx so it is rank 1 and I realized that for any cast form of weight 2 would be r and if it's galore representation is induced representation of a maximum idea then this guy modulo or the prime corresponding to this f doesn't vanish because this is concentrated at the weight 1 okay so this f is locally decomposed you can do also imaginary quadratic version and one Eriksson and Francesca Tia told me that they also did that and so this kind of result is not so hard anyway it's basically independent of what I said but this one I need for my book okay so this tells you this is almost that a year earlier a couple of years earlier Bettina Adam found a criterion so you consider now iron the weight space of rigid analytic weight space you have an iron curve of form of measure and you have a point say of this induced representation of phi after representation and the day this guy studied the Lamecation Index and he had a criterion to show the Lamecation Index is 2 but his criterion is very hard to verify criterion is a nice criterion but I don't think he verified in any case but anyway I'll tell you what it is so you take you take phi minus and then you take the maximal Lamecation and Lamefied outside and P Vax Pnil and Lamefied outside P and the same thing P sigma you make a composite and then you have inside the Zp multiple Zp it's coming out here Zp to the sum power s so I have this phi minus and infinity then he considered infinity everywhere and Lamefied P extension so you have a big Galois group X and you have Q so you can make a portion so to speak tile orientation portion that gives rise to some field paths here and he wanted to have this line to be to be a V over essentially L5 minus then he is two he uses the not pseudo-character but pseudo-representation but pseudo-character of T1 to T4 unique T4 is equivalent to virus pseudo-representation and I know that this raw T restricted H is universal Kay Lee Hamilton so every virus pseudo-character also factors through trace of this guy therefore I he his staff just a computation of tangent vector at this weight space its weight one space and you over this and he competes tangent space of this and why pseudo-representation theory space collapse so that is the localization if it takes my localization is just a W localized that's what is proved under his criteria but I could change everything here to something inside of so I have an infinity and an infinity kind of since I can basically intercepted with every raw T because all the pseudo-representation factors through this guy and the gamma group of raw T over gamma group of a flow bar same as in the L2 of T and up to a very young factor you forget about it you can intercept it in a certain two T so I take this is G prime and I just take the intersection this one and that is I call it G then there's a theory by Richard Pink he classified all this is p-provider it is O by flow bar he classified all p-provider crosses some group of this one and I know in Asia basically have unipotent guy really big so I apply so I don't actually take really an intersection but I need to modify Bettina's argument quite but anyway I got idea from him so what I do I have five minutes left so what I do is to consider this everything I do it over M over T and inside I have M infinity and say phi minus this is max multiple zp extension inside and corresponding to infinity the max every phi of the alien extension and I can determine the Galois group very explicitly all of them and then I do his business I get he is true and that's the end of the story I might not in case it's more technical and because it has a very big C.M. quotient and it's not the epsilon kind of small so it's more technical but it's also you can do basically this type of argument I have a corresponding result but anyway if I have some opportunity I'll explain it wasn't in case but the real case is easier but some is almost consumed one now that's perhaps sufficient thank you